The only toss-up in this scenario is Batman vs Cap. Cap would win in certain situations because he's got augmented strength and agility, so like if they were both dropped into a UFC fight with just their shorts on, Cap might win even though Batman isn't above fighting dirty. But in a situation where Batman can use stealth and his arsenal, all Cap has is good looks and a shield. I think Batman might take him.

Otherwise, the League beats the Avengers in every fight except possibly for Wonder Woman, Thor is probably stronger plus the whole lightning thing. She only wins if he objects to hitting a lady, and she can use the lasso of truth to get him to simmer down.

However, there's a way to stack things so the Avengers win:

Hulk vs Flash -- he only needs one punch.
Thor vs Green Lantern -- I don't think the Lantern can circumvent Asgardian god magic.
Ironman vs Batman
Then you throw Black Widow at Wonder Woman for a gimme, because you're not beating her, and then throw Cap at Superman. Either Supes wins in 0.0001 seconds, or Cap talks him out of fighting altogether since they both stand for Truth, Justice and the American Way. So, the Avengers squeak out a victory 3 to 2 or 4 to 1, when the traditional fights would see them get their clocks cleaned. It's all about tactics.

The X-men have some unfair advantages with the telepaths, I don't think they're easy to stop. However, I can't think of anyone on the X-men who could otherwise circumvent Supes or GL because let's face it, how do you beat Supes? And a ring that let's you do anything you can imagine is tough to fight. Maybe Storm could switch fights, and the Colossus could go toe-to-toe with Wonder Woman, but I think she'd take him the same way I suggested she take Thor. Storm has range.

Wolverine and Batman both have ninja training, but Wolverine has an unbreakable skeleton and a healing factor. Sorry Bats.

X-Men versus Avengers

X-men hands down. I never liked the Avengers very much -- they're the obvious Marvel "All-star" team, stacked with the heavy-hitters. I don't even understand why they're a team, other than the fact they're the biggest characters. The Fantastic Four is centred around a family, and the X-men become a family because they're a disadvantaged minority that's persecuted against. They actually stand for important social issues, like integration and equality. Writers on the X-men get to grapple with societal issues like gender, racism and intolerance without being too overtly political. What do the Avengers do besides fight Loki and Kang?